Polynomial alias higher degree fuzzy transform of complex-valued functions

Abstract

In this article, we propose a general approach to the computation of components of the direct higher degree fuzzy transform. Apart from the orthogonal bases of the subspaces of polynomials of weighted Hilbert spaces with respect to a generalized uniform fuzzy partition, which are used in all papers on fuzzy transform of higher degree, we admit also the non-orthogonal bases. An advantage of using non-orthogonal bases consists in the possibility of replacing orthogonal polynomials, derivation of which by the Gram–Schmidt orthogonalization process can be questionable difficult or imprecise, by suitable non-orthogonal polynomials of much simpler form. We present a simple matrix calculus and show how it can be used to introduce the components of the direct higher degree fuzzy transform. With the help of the monomial basis, we prove a convergence theorem and an approximation theorem for the higher degree fuzzy transform. The results are illustrated by examples including a comparison with standard methods.